Symmetric polynomial
A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a symmetric function in its variables, that is, is invariant under all permutations of the variables:
\[ \tag{1} f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)). \]
The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
The most important examples of symmetric polynomials are the elementary symmetric polynomials
\[ s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k} \]
and the power sums
\[ p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k. \]
The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:
\[ \begin{aligned} p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{k-1} p_1 s_{k-1}+ (-1)^{k} k s_{k} &= 0 \quad &\text{if }1 \leq k \leq n, \\ p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{n-1} p_{k-n+1} s_{n-1}+ (-1)^{n} p_{k-n} s_{n} &= 0 \quad &\text{if }k > n. \end{aligned} \]
For the elementary symmetric polynomials $s_1,\ldots,s_k$ ($1 \leq k \leq n$) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, $x^n+\alpha_1 x^{n-1}+ \ldots + \alpha_n$, one has $\alpha_k=(-1)^k s_k$ (see Viète theorem).
The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra $S(x_1,\ldots,x_n)$. If the field has characteristic 0, then the polynomials $p_1,\ldots,p_n$ also form a set of free generators of this algebra.
A skew-symmetric, or alternating, polynomial is a polynomial $f(x_1,\ldots,x_n)$ satisfying the relation (1) if $\pi$ is even and the relation
\[ f(x_1,\ldots,x_n) = -f(\pi(x_1),\ldots,\pi(x_n)) \]
if $\pi$ is odd. Any skew-symmetric polynomial can be written [Pr, Thm. 3.1.2] in the form $\Delta_n g$, where $g$ is a symmetric polynomial and
\[ \Delta_n = \prod_{i<j} (x_i-x_j). \]
This representation is not unique, in view of the relation $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.
References
[Ku] | A.G. Kurosh, "Higher algebra", MIR (1972) (Translated from Russian) |
[Ko] | A.I. Kostrikin, "Introduction to algebra", Springer (1982) (Translated from Russian) |
[MP] | A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra", Pergamon (1965) (Translated from Russian) |
[Pr] | V. V. Prasolov, "Polynomials", vol. 11 of "Algorithms and Computation in Mathematics", Springer (2004). |
Comments
Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials ($S$-functions) $S_{\{\lambda\}}$. These are defined for any partition $\{\lambda_1,\ldots,\lambda_p\}=\lambda$, and include as special cases the above functions, e.g. $S_{\{1,\ldots,1\}}=s_k$, $S_{\{ k \}}=p_k$ (see, e.g., [Li, Chapt. VI]).
In general, the discriminant of the polynomial $a_0 x^n + \ldots + a_n $ with roots $x_1,\ldots,x_n$ is defined as $D=a_0^{2n-2} \prod_{1 \leq i < j \leq n} (x_i-x_j)^2$, and satisfies
\[ D = a_0^{2n-2} \begin{vmatrix} p_0 & \ldots & p_{n-1} \\ p_1 &\ldots & p_n \\ \ldots & \ldots & \ldots \\ p_{n-1} & \ldots & p_{2n-2} \end{vmatrix}, \]
with $p_0(x_1,\ldots,x_n)=n$.
See Discriminant.
Let $A_n \subset S_n$ be the alternating group, consisting of the even permutations. The ring of polynomials $k[x_1,\ldots,x_n]^{A_n}$ of polynomials over a field $k$ obviously contains the elementary symmetric functions $s_1,\ldots,s_n$ and $\Delta_n=\prod_{i<j} (x_i-x_j)$. If $k$ is not of characteristic $2$, the ring of polynomials is generated by $s_1,\ldots,s_n$ and $\Delta$, and the ideal of relations is generated by $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$. The condition $\operatorname{char} k \neq 2$ is also necessary for the statement that every skew-symmetric polynomial is of the form $\Delta_n g$ with $g$ symmetric. More precisely, what is needed for this is that $2u=0$ implies $u=0$ for $u \in k$.
References
[Ja] | N. Jacobson, "Basic algebra" , 1, Freeman (1974) |
[Ku2] | A.G. Kurosh, "An introduction to algebra" , MIR (1971) (Translated from Russian) |
[vdW] | B.L. van der Waerden, "Algebra" , 1, Springer (1967) (Translated from German) |
[Li] | D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950) |
[Po] | V. Poénaru, "Singularités $\mathcal{C}^\infty$ en présence de symmétrie" , Springer (1976), pp. 14ff |
[Co] | P.M. Cohn, "Algebra" , 1, Wiley (1982) pp. 181 |
Symmetric polynomial. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Symmetric_polynomial&oldid=39117